Definition:Positive Definite

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Positive Definite may refer to:

Positive Definite Function on Ring

Let $\struct {R, +, \times}$ be a ring whose zero is denoted $0_R$.

Let $f: R \to \R$ be a (real-valued) function on $R$.

Then $f$ is positive definite if and only if:

$\forall x \in R: \begin {cases} \map f x = 0 & : x = 0_R \\ \map f x > 0 & : x \ne 0_R \end {cases}$

Positive Definite Matrix

Let $\mathbf A$ be a symmetric square matrix of order $n$.

$\mathbf A$ is positive definite if and only if:

all the eigenvalues of $\mathbf A$ are strictly positive.

Positive Definite Vector Space

Let $\mathbf V$ be a vector space such that:

$\forall \mathbf v \in \mathbf V: \mathbf v \cdot \mathbf v > 0$

where $\mathbf v \cdot \mathbf v$ denotes the dot product.

Then $\mathbf V$ is a positive definite vector space.

This article is complete as far as it goes, but it could do with expansion. In particular: Several more instances of this in the various contexts, in particular for hermitian matrices. Research to be done. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Definition:Positive
• Definition:Definite

• Results about positive definite can be found here.